The fact of the matter is that three different enemies have to be eliminated to complete this quest. That might be challenging with the servers being overly populated and unstable as of this moment. But it can be done, as soon as you know the exact location of each enemy. Here is a complete “New Initiate, Old Order” walkthrough guide!
New Initiate, Old Order Guide – New World
This quest can be started by talking with Yonas The Hermit at the Fisherman’s Bend. You’ll need to prove to him that you have what it takes to become a Soulwarden, by eliminating all enemies at Saircor Ruins.
After you have the quest, you’ll need to make your way south, to the First Light’s Settlement. However, when you get there, go east to reach the following location:

The bridge is where the Soulwarden Trials take place. Basically, once you get here, the objective will turn to eliminating XY number of a couple of particular enemies.
Where most players fail is they get off the bridge to look for the enemies, because they do have a really small spawn rate. But they do spawn, so wait for those enemies on the bridge.
Depending on where you’re standing on the bridge, the start, the middle, the exit, there will be different enemies that spawn. Go through them one by one and you shouldn’t have any issues. But make sure you stay close to their spawns and damage them enough to get them to count towards the objective.